Soil Behavior: Dynamic Soil-Structure Interactions - Dynamic Behavior of Concrete and Seismic Engineering - page 157

Unlike local boundaries, consistent boundaries couple all the degrees of freedom
of the boundary nodes and perfectly absorb all kinds of waves. Formulating these
boundaries ([KAU 74], [LYS 72] and [WAA 72]) involves frequency dependent
terms; therefore, they can only be used for resolutions in the frequency field. There
are no consistent boundaries to represent the effect of the semi-space underlying the
model. This is why the lower boundary of the model is supposed to be rigid. If it is
chosen to be deep enough (about a structure wide), the reverberation phenomena on
this boundary become negligible. Actually, the field of the waves reverberated by
the structure mainly consists of surface waves which die down fast with depth.
Besides, we can take advantage of this property by having a moving lower
boundary, with a variable meshing, which is deeper the lower the studied
frequencies. In fact, low frequencies die down more slowly but they require coarser
meshing to ensure a correct transmission of the wave [LYS 81].
4.3.4.4. Digital integration pattern
It is theoretically possible to choose a digital integration pattern that is either a
time, a frequency or a mode pattern. These aspects will not be explained in detail
below; for more information, see [PEC 84]. We will only mention that for linear
problems, the frequency integration pattern is best suited to solving soil-structure
interaction problems because of the formulation of the soil behavior law (section
4.2), the dependence on frequency of the impedance matrices used in sub-structure
methods, and the absorbing boundaries used to estimate the kinematic interaction
displacements or for global method resolutions.
However, if a time or mode integration pattern is preferred, the variation of
impedances with frequencies has to be taken into account. This can be implemented
either by successive iterations allowing us to adjust the impedance to the frequency
of the soil-structure interaction mode, or by resorting to analog models using
springs, dashpots and additional mass (Figure 4.15).
Such models, which originate in cone models [WOL 95], lead to very satisfying
approximations of the impedance functions [PEC 94]. Nevertheless, due to the
presence of an additional mass, they require the use of precautions related to the
definition of the actual movement of the foundations. They can be equally used for
either mode analysis or time integration.